We discuss modifications in the integral representation of the Riemann zeta-function that lead to generalizations of the Riemann functional equation that preserves the symmetry s → (1 − s ) in the critical strip. By modifying one integral representation of the zeta-function with a cut-off that does exhibit the symmetry x ↦ 1/ x , we obtain a generalized functional equation involving Bessel functions of second kind. Next, with another cut-off that does exhibit the same symmetry, we obtain a generalization for the functional equation involving only one Bessel function of second kind. Some connection between one regularized zeta-function and the Laplace transform of the heat kernel for the Euclidean and hyperbolic space is discussed.

中文翻译：

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正则化zeta函数和扩散过程的函数方程

我们讨论了对黎曼zeta函数积分表示的修改，这些变换导致对黎曼函数方程的推广，该方程在临界带中保留了对称s→（1- s）。通过用确实表现出对称性x↦1 / x的截止点修改zeta函数的一个积分表示，我们获得了包含第二类Bessel函数的广义函数方程。接下来，通过另一个确实表现出相同对称性的截断，我们获得了仅包含第二种贝塞尔函数的泛函的泛化方程。讨论了一个正则zeta函数与欧氏空间和双曲线空间的热核的拉普拉斯变换之间的某些联系。